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A005846 Primes of the form n^2 + n + 41.
(Formerly M5273)
96
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Note that 41 is the largest of Euler's Lucky numbers (A014556). - Lekraj Beedassy, Apr 22 2004

a(n)=A117530(13,n) for n<=13: a(1)=A117530(13,1)=A014556(6)=41, A117531(13)=13. - Reinhard Zumkeller, Mar 26 2006

The link to E. Wegrzynowski contains the following incorrect statement: "It is possible to find a polynomial of the form n^2 + n + B that gives prime numbers for n = 0...A, A being any number." It is known that the maximum is A = 39 for B = 41. - Luis Rodriguez (luiroto(AT)yahoo.com), Jun 22 2008

Contrary to the last comment, Mollin's Theorem 2.1 shows that any A is possible if the Prime k-tuples Conjecture is assumed. - T. D. Noe, Aug 31 2009

a(n) can be generated by a recurrence based on the gcd in the type of Eric Rowland and Aldrich Stevens. See the recurrence in PARI under PROG. - Mike Winkler, Oct 02 2013

REFERENCES

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Zak Seidov, Table of n, a(n) for n = 1..10000.

Phil Carmody, Drag Racing Prime Numbers! [from Vladimir Joseph Stephan Orlovsky, Jul 24 2011]

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

R. A. Mollin, Prime-producing quadratics, Amer. Math. Monthly 104 (1997), 529-544.

E. Wegrzynowski, Les formules simples qui donnent des nombres premiers en grande quantite

Eric Weisstein's World of Mathematics, Euler Prime

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

FORMULA

a(n) = A056561(n)^2 + A056561(n) + 41.

EXAMPLE

a(39) = 1601 = 39^2 + 39 + 41 is in the sequence because it is prime.

1681 = 40^2 + 40 + 41 is not in the sequence because 1681 = 41*41.

MAPLE

for y from 0 to 10 do

U := y^2+y+41;

if isprime(U) = true then print(U) end if ;

end do:

# Matt C. Anderson, Jan 04 2013

MATHEMATICA

Select[Table[n^2 + n + 41, {n, 0, 59}], PrimeQ] (* Alonso del Arte, Dec 08 2011 *)

PROG

(PARI) for(n=1, 1e3, if(isprime(k=n^2+n+41), print1(k", "))) \\ Charles R Greathouse IV, Jul 25 2011

(Haskell)

a005846 n = a005846_list !! (n-1)

a005846_list = filter ((== 1) . a010051) a202018_list

-- Reinhard Zumkeller, Dec 09 2011

(PARI) {k=2; n=1; for(x=1, 100000, f=x^2+x+41; g=x^2+3*x+43; a=gcd(f, g-k); if(a>1, k=k+2); if(a==x+2-k/2, print(n" "a); n++))} \\ Mike Winkler, Oct 02 2013

CROSSREFS

Cf. A048988, A007634, A056561, A002378, A007635.

Intersection of A000040 and A202018; A010051.

Sequence in context: A257362 A155884 A202018 * A273756 A154498 A223458

Adjacent sequences:  A005843 A005844 A005845 * A005847 A005848 A005849

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Henry Bottomley, Jun 26 2000

STATUS

approved

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Last modified September 21 10:08 EDT 2017. Contains 292297 sequences.